ar X iv : h ep - p h / 99 06 46 3 v 1 2 3 Ju n 19 99 UAB – FT – 468 IFT – P . 053 /

Neutrinos mediate long range forces among macroscopic bodies in vacuum. When the bodies are placed in the neutrino cosmic background, these forces are modified. Indeed, at distances long compared to the scale T −1 , the relic neutrinos completely screen off the 2-neutrino exchange force, whereas for small distances the interaction remains unaffected. Dispersion potentials arising from double particle exchange have been systematically studied in a wide variety of physical contexts and with quite different scopes and purposes [1]. Indeed, the studies include pure QED phenomena such as Van der Waals interactions [2], two neutrino forces among macroscopic bodies [3], forces mediated by scalar particles [4, 5] found in recent completions of the standard model (e.g. superlight scalar partners of the gravitino), etc. In particular 2-neutrino exchange forces have been repeatedly scrutinized since first discussed by Feinberg and Sucher. An aspect that has been reanalysed in recent work [6] is the observation raised in [7] that the cosmic neutrino heat bath has an effect on long range neu-trino interactions. In both these papers [7, 6] an approximate neutrino distribution function was used that simplified the calculations. The claim was that for small neutrino chemical potential, the background neutrinos can be considered nearly Boltzmann distributed and this fact, while only introduces a small distortion into the long range forces, makes the calculations much easier. But the actual phase-space distribution for relic cosmological neutrinos has a Fermi-Dirac shape. Indeed, any fermionic species in thermal equilibrium which at time t D and temperature T D of decoupling was highly relativistic followed an equilibrium distribution n(p, t D) = [exp (E/T D) + 1] −1. After decoupling, the energy is red shifted by the expansion of the Universe, E(t) = E(t D) (R(t D)/R(t)), as the number density decreases like R −3. As a result, the phase-space distribution at time t will keep the Fermi-Dirac form with the temperature T (t) = T D (R(t D)/R(t)). In the present paper we use the exact Fermi-Dirac neutrino distribution function with arbitrary chemical potential and observe that the long distance results are drastically modified even for small chemical potential in contrast to previous claims. We neglect the effect of a neutrino mass which for the phenomenologically suggested values would not affect the present results. We comment in passing that there has been in the recent literature [8] renewed interest in cosmic neutrino degeneracy which could make …